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- 436-431 MECHANICS 4 UNIT 2 MECHANICAL VIBRATION J.M. KRODKIEWSKI 2008 THE UNIVERSITY OF MELBOURNE Department of Mechanical and Manufacturing Engineering . 1
- 2 MECHANICAL VIBRATIONS Copyright C 2008 by J.M. Krodkiewski The University of Melbourne Department of Mechanical and Manufacturing Engineering
- CONTENTS 0.1 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I MODELLING AND ANALYSIS 7 1 MECHANICAL VIBRATION OF ONE-DEGREE-OF-FREEDOM LINEAR SYSTEMS 9 1.1 MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM . . . . 9 1.1.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2 ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM . . . . . . 28 1.2.1 Free vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.2 Forced vibration . . . . . . . . . . . . . . . . . . . . . . . . 34 1.2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2 MECHANICAL VIBRATION OF MULTI-DEGREE-OF-FREEDOM LINEAR SYSTEMS 66 2.1 MODELLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.1.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.1.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . 67 2.1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.2 ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM . . . . . 93 2.2.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.2.2 Modal analysis - case of small damping . . . . . . . . . . 102 2.2.3 Kinetic and potential energy functions - Dissipation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.3 ENGINEERING APPLICATIONS . . . . . . . . . . . . . . . . . . . 151 2.3.1 Balancing of rotors . . . . . . . . . . . . . . . . . . . . . . 151 2.3.2 Dynamic absorber of vibrations . . . . . . . . . . . . . . 157 3 VIBRATION OF CONTINUOUS SYSTEMS 162 3.1 MODELLING OF CONTINUOUS SYSTEMS . . . . . . . . . . . . . 162 3.1.1 Modelling of strings, rods and shafts . . . . . . . . . . . 162 3.1.2 Modelling of beams . . . . . . . . . . . . . . . . . . . . . . 166
- CONTENTS 4 3.2 ANALYSIS OF CONTINUOUS SYSTEMS . . . . . . . . . . . . . . 168 3.2.1 Free vibration of strings, rods and shafts . . . . . . . . . 168 3.2.2 Free vibrations of beams . . . . . . . . . . . . . . . . . . . 174 3.2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 3.3 DISCRETE MODEL OF THE FREE-FREE BEAMS . . . . . . . . . 214 3.3.1 Rigid Elements Method. . . . . . . . . . . . . . . . . . . . 214 3.3.2 Finite Elements Method. . . . . . . . . . . . . . . . . . . . 217 3.4 BOUNDARY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . 225 3.5 CONDENSATION OF THE DISCREET SYSTEMS . . . . . . . . . 226 3.5.1 Condensation of the inertia matrix. . . . . . . . . . . . . 227 3.5.2 Condensation of the damping matrix. . . . . . . . . . . . 228 3.5.3 Condensation of the stiﬀness matrix. . . . . . . . . . . . 228 3.5.4 Condensation of the external forces. . . . . . . . . . . . . 228 3.6 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 II EXPERIMENTAL INVESTIGATION 237 4 MODAL ANALYSIS OF A SYSTEM WITH 3 DEGREES OF FREE- DOM 238 4.1 DESCRIPTION OF THE LABORATORY INSTALLATION . . . . . 238 4.2 MODELLING OF THE OBJECT . . . . . . . . . . . . . . . . . . . . 239 4.2.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . 240 4.3 ANALYSIS OF THE MATHEMATICAL MODEL . . . . . . . . . . . 241 4.3.1 Natural frequencies and natural modes of the undamped system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 4.3.2 Equations of motion in terms of the normal coordinates - transfer functions . . . . . . . . . . . . . . . . . . . . . . 241 4.3.3 Extraction of the natural frequencies and the natural modes from the transfer functions . . . . . . . . . . . . . 242 4.4 EXPERIMENTAL INVESTIGATION . . . . . . . . . . . . . . . . . 243 4.4.1 Acquiring of the physical model initial parameters . . 243 4.4.2 Measurements of the transfer functions . . . . . . . . . . 244 4.4.3 Identiﬁcation of the physical model parameters . . . . 245 4.5 WORKSHEET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
- INTRODUCTION. 5 0.1 INTRODUCTION. The purpose of this text is to provide the students with the theoretical background and engineering applications of the theory of vibrations of mechanical systems. It is divided into two parts. Part one, Modelling and Analysis, is devoted to this solu- tion of these engineering problems that can be approximated by means of the linear models. The second part, Experimental Investigation, describes the laboratory work recommended for this course. Part one consists of four chapters. The ﬁrst chapter, Mechanical Vibration of One-Degree-Of-Freedom Linear System, illustrates modelling and analysis of these engineering problems that can be approximated by means of the one degree of freedom system. Infor- mation included in this chapter, as a part of the second year subject Mechanics 1, where already conveyed to the students and are not to be lectured during this course. However, since this knowledge is essential for a proper understanding of the following material, students should study it in their own time. Chapter two is devoted to modeling and analysis of these mechanical systems that can be approximated by means of the Multi-Degree-Of-Freedom models. The Newton’s-Euler’s approach, Lagrange’s equations and the inﬂuence coeﬃcients method are utilized for the purpose of creation of the mathematical model. The considerations are limited to the linear system only. In the general case of damping the process of looking for the natural frequencies and the system forced response is provided. Application of the modal analysis to the case of the small structural damping results in solution of the initial problem and the forced response. Dynamic balancing of the rotating elements and the passive control of vibrations by means of the dynamic absorber of vibrations illustrate application of the theory presented to the engineering problems. Chapter three, Vibration of Continuous Systems, is concerned with the problems of vibration associated with one-dimensional continuous systems such as string, rods, shafts, and beams. The natural frequencies and the natural modes are used for the exact solutions of the free and forced vibrations. This chapter forms a base for development of discretization methods presented in the next chapter In chapter four, Approximation of the Continuous Systems by Dis- crete Models, two the most important, for engineering applications, methods of approximation of the continuous systems by the discrete models are presented. The Rigid Element Method and the Final Element Method are explained and utilized to produce the inertia and stiﬀness matrices of the free-free beam. Employment of these matrices to the solution of the engineering problems is demonstrated on a number of examples. The presented condensation techniques allow to keep size of the discrete mathematical model on a reasonably low level. Each chapter is supplied with several engineering problems. Solution to some of them are provided. Solution to the other problems should be produced by students during tutorials and in their own time. Part two gives the theoretical background and description of the laboratory experiments. One of them is devoted to the experimental determination of the nat- ural modes and the corresponding natural frequencies of a Multi-Degree-Of-Freedom-
- INTRODUCTION. 6 System. The other demonstrates the balancing techniques.
- Part I MODELLING AND ANALYSIS 7
- 8 Modelling is the part of solution of an engineering problems that aims to- wards producing its mathematical description. This mathematical description can be obtained by taking advantage of the known laws of physics. These laws can not be directly applied to the real system. Therefore it is necessary to introduce many assumptions that simplify the engineering problems to such extend that the physic laws may be applied. This part of modelling is called creation of the physical model. Application of the physics law to the physical model yields the wanted mathematical description that is called mathematical model. Process of solving of the mathematical model is called analysis and yields solution to the problem considered. One of the most frequently encounter in engineering type of motion is the oscillatory motion of a mechanical system about its equilibrium position. Such a type of motion is called vibration. This part deals with study of linear vibrations of mechanical system.
- Chapter 1 MECHANICAL VIBRATION OF ONE-DEGREE-OF-FREEDOM LINEAR SYSTEMS DEFINITION: Any oscillatory motion of a mechanical system about its equilibrium position is called vibration. 1.1 MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM DEFINITION: Modelling is the part of solution of an engineering problem that aims for producing its mathematical description. The mathematical description of the engineering problem one can obtain by taking advantage of the known lows of physics. These lows can not be directly applied to the real system. Therefore it is necessary to introduce many assumptions that simplify the problem to such an extend that the physic laws may by apply. This part of modelling is called creation of the physical model. Application of the physics law to the physical model yields the wanted mathematical description which is called mathematical model. 1.1.1 Physical model As an example of vibration let us consider the vertical motion of the body 1 suspended on the rod 2 shown in Fig. 1. If the body is forced out from its equilibrium position and then it is released, each point of the system performs an independent oscillatory motion. Therefore, in general, one has to introduce an inﬁnite number of independent coordinates xi to determine uniquely its motion. t xi i 1 2 Figure 1
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 10 DEFINITION: The number of independent coordinates one has to use to determine the position of a mechanical system is called number of degrees of freedom According to this deﬁnition each real system has an inﬁnite number of degrees of freedom. Adaptation of certain assumptions, in many cases, may results in reduc- tion of this number of degrees of freedom. For example, if one assume that the rod 2 is massless and the body 1 is rigid, only one coordinate is suﬃcient to determine uniquely the whole system. The displacement x of the rigid body 1 can be chosen as the independent coordinate (see Fig. 2). t x i x ix 1 2 Figure 2 Position xi of all the other points of our system depends on x. If the rod is uniform, its instantaneous position as a function of x is shown in Fig. 2. The following analysis will be restricted to system with one degree of freedom only. To produce the equation of the vibration of the body 1, one has to produce its free body diagram. In the case considered the free body diagram is shown in Fig. 3. t x 1 R G Figure 3 The gravity force is denoted by G whereas the force R represents so called restoring force. In a general case, the restoring force R is a non-linear function of
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 11 the displacement x and the instantaneous velocity ˙x of the body 1 (R = R(x, ˙x)). The relationship between the restoring force R and the elongation x as well as the velocity ˙x is shown in Fig. 4a) and b) respectively. x x . R R 0 0 a) b) Figure 4 If it is possible to limit the consideration to vibration within a small vicinity of the system equilibrium position, the non-linear relationship, shown in Fig. 4 can be linearized. R=R(x, ˙x) ≈ kx + c ˙x (1.1) The ﬁrst term represents the system elasticity and the second one reﬂects the system’s ability for dissipation of energy. k is called stiﬀness and c is called coeﬃcient of damping. The future analysis will be limited to cases for which such a linearization is acceptable form the engineering point of view. Such cases usually are refer to as linear vibration and the system considered is call linear system. Result of this part of modelling is called physical model. The physical model that reﬂects all the above mention assumption is called one-degree-of-freedom linear system. For presentation of the physical model we use symbols shown in the Fig. 5.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 12 m ϕ x m,I m k A, J, E ϕ k c ϕ .c rigid block of mass rigid body of mass m m and moment of inertia I particle of of mass m massless spring of stifness k massless beam area second moment of area and Young modulusA, J E (linear motion) (angular motion) (linear motion) massless spring of stifness k (angular motion) massless damper of damping coefficient c (linear motion) massless damper of damping coefficient c (angular motion) Figure 5 1.1.2 Mathematical model To analyze motion of a system it is necessary to develop a mathematical description that approximates its dynamic behavior. This mathematical description is referred to as the mathematical model. This mathematical model can be obtained by application of the known physic lows to the adopted physical model. The creation of the phys- ical model, has been explained in the previous section. In this section principle of producing of the mathematical model for the one-degree-of-freedom system is shown. Let us consider system shown in Fig. 6.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 13 k m c x mg sk x Figure 6 Let as assume that the system is in an equilibrium. To develop the mathe- matical model we take advantage of Newton’s generalized equations. This require introduction of the absolute system of coordinates. In this chapter we are assuming that the origin of the absolute system of coordinates coincides with the centre of gravity of the body while the body stays at its equilibrium position as shown in Fig. 6. The resultant force of all static forces (in the example considered gravity force mg and interaction force due to the static elongation of spring kxs) is equal to zero. Therefore, these forces do not have to be included in the Newton’s equations. If the system is out of the equilibrium position (see Fig. 7) by a distance x, there is an increment in the interaction force between the spring and the block. This increment is called restoring force. k m c x mg sk x -kx=-kx x>0 x<0 kx=-kx Figure 7 In our case the magnitude of the restoring force is |FR| = k |x| If x > 0, the restoring force is opposite to the positive direction of axis x. Hence FR = −k |x| = −kx If x < 0, the restoring force has the same direction as axis x. Hence FR = +k |x| = −kx Therefore the restoring force always can be represented in the equation of motion by term FR = −kx (1.2)
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 14 k m c x mg sk x -cx=-cx x>0 x<0 . . . +cx=-cx . . . Figure 8 Creating the equation of motion one has to take into consideration the interac- tion force between the damper and the block considered (see Fig. 8). This interaction force is called damping force and its absolute value is |FD| = c | ˙x| . A very similar to the above consideration leads to conclusion that the damping force can be represented in the equation of motion by the following term FD = −c ˙x (1.3) k m c x F (t)ex Figure 9 The assumption that the system is linear allows to apply the superposition rules and add these forces together with the external force Fex(t) (see Fig. 9). Hence, the equation of motion of the block of mass m is m¨x = −kx − c ˙x + Fex(t) (1.4) Transformation of the above equation into the standard form yields ¨x + 2ςωn ˙x + ω2 nx = f(t) (1.5) where
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 15 ωn = r k m ; 2ςωn = c m ; f(t) = Fex(t) m (1.6) ωn - is called natural frequency of the undamped system ς - is called damping factor or damping ratio f(t) - is called unit external excitation The equation 1.5 is known as the mathematical model of the linear vibration of the one-degree-of-freedom system.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 16 1.1.3 Problems Problem 1 A c y k1 k2 m Figure 10 The block of mass m (see Fig. 10)is restricted to move along the vertical axis. It is supported by the spring of stiﬀness k1, the spring of stiﬀness k2 and the damper of damping coeﬃcient c. The upper end of the spring k2 moves along the inertial axis y and its motion is governed by the following equation yA = a sin ωt were a is the amplitude of motion and ω is its angular frequency. Produce the equation of motion of the block.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 17 Solution A c y k1 k2 m x Figure 11 Let us introduce the inertial axis x in such a way that its origin coincides with the centre of gravity of the block 1 when the system is in its equilibrium position (see Fig. 11. Application of the Newton’s low results in the following equation of motion m¨x = −k2x − k1x + k2y − c ˙x (1.7) Its standard form is ¨x + 2ςωn ˙x + ω2 nx = q sin ωt (1.8) where ω2 n = k1 + k2 m 2ςωn = c m q = k2a m (1.9)
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 18 Problem 2 r R 1 2 Figure 12 The cylinder 1 (see Fig. 12) of mass m and radius r is plunged into a liquid of density d. The cylindric container 2 has a radius R. Produce the formula for the period of the vertical oscillation of the cylinder.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 19 Solution r R x x G z V V 1 2 Figure 13 Let us introduce the inertial axis x in such a way that its origin coincides with the centre of gravity of the cylinder 1 when the system is in its equilibrium position (see Fig. 13. If the cylinder is displaced from its equilibrium position by a distance x, the hydrostatic force acting on the cylinder is reduced by ∆H = (x + z) dgπr2 (1.10) Since the volume V1 must be equal to the volume V2 we have V1 = πr2 x = V2 = π ¡ R2 − r2 ¢ z (1.11) Therefore z = r2 R2 − r2 x (1.12) Introducing the above relationship into the formula 1.10 one can get that ∆H = µ x + r2 R2 − r2 x ¶ dgπr2 = πdg µ R2 r2 R2 − r2 ¶ x (1.13) According to the Newton’s law we have m¨x = −dgπ µ R2 r2 R2 − r2 ¶ x (1.14) The standard form of this equation of motion is ¨x + ω2 nx = 0 (1.15) where ω2 n = dg m π µ R2 r2 R2 − r2 ¶ (1.16) The period of the free oscillation of the cylinder is Tn = 2π ωn = 2π Rr s m (R2 − r2) πdg = 2 Rr s mπ (R2 − r2) dg (1.17)
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 20 Problem 3 c R G D L m 1 Figure 14 The disk 1 of mass m and radius R (see Fig. 14) is supported by an elastic shaft of diameter D and length L. The elastic properties of the shaft are determined by the shear modulus G. The disk can oscillate about the vertical axis and the damping is modelled by the linear damper of a damping coeﬃcient c. Produce equation of motion of the disk
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 21 Solution c R G D L m 1 ϕ Figure 15 Motion of the disk is governed by the generalized Newton’s equation I ¨ϕ = −ksϕ − cR2 ˙ϕ (1.18) where I = mR2 2 - the moment of inertia of the disk ks = T ϕ = T T L JG = JG L = πD4G 32L the stiﬀness of the rod Introduction of the above expressions into the equation 1.18 yields I ¨ϕ + cR2 ˙ϕ + πD4 G 32L ϕ = 0 (1.19) or ¨ϕ + 2ςωn ˙ϕ + ω2 nϕ = 0 (1.20) where ω2 n = πD4 G 32LI 2ςωn = cR2 I (1.21)
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 22 Problem 4 O k c l b a 1 m Figure 16 The thin and uniform plate 1 of mass m (see Fig. 16) can rotate about the horizontal axis O. The spring of stiﬀness k keeps it in the horizontal position. The damping coeﬃcient c reﬂects dissipation of energy of the system. Produce the equation of motion of the plate.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 23 Solution O k c l b a 1 m ϕ Figure 17 Motion of the plate along the coordinate ϕ (see Fig. 17) is govern by the generalized Newton’s equation I ¨ϕ = M (1.22) The moment of inertia of the plate 1 about its axis of rotation is I = mb2 6 (1.23) The moment which act on the plate due to the interaction with the spring k and the damper c is M = −kl2 ϕ − cb2 ˙ϕ (1.24) Hence mb2 6 ¨ϕ + kl2 ϕ + cb2 ˙ϕ = 0 (1.25) or ¨ϕ + 2ςωn ˙ϕ + ω2 nϕ = 0 (1.26) where ω2 n = 6kl2 mb2 2ςωn = 6c m (1.27)
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 24 Problem 5 E,I M m ω t l c µ Figure 18 The electric motor of mass M (see Fig. 18)is mounted on the massless beam of length l, the second moment of inertia of its cross-section I and the Young modulus E. The shaft of the motor has a mass m and rotates with the angular velocity ω. Its unbalance (the distance between the axis of rotation and the shaft centre of gravity) is µ. The damping properties of the system are modelled by the linear damping of the damping coeﬃcient c. Produce the equation of motion of the system.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 25 Problem 6 yA 0 c k d D l L Figure 19 The wheel shown in the Fig. 19 is made of the material of a density . It can oscillate about the horizontal axis O. The wheel is supported by the spring of stiﬀness k and the damper of the damping coeﬃcient c. The right hand end of the damper moves along the horizontal axis y and its motion is given by the following equation y = a sin ωt Produce the equation of motion of the system
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 26 Problem 7 L r 1 2 Figure 20 The cylinder 1 of mass m is attached to the rigid and massless rod 2 to form the pendulum shown in the Fig. 20. Produce the formula for the period of oscillation of the pendulum.
- MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM 27 Problem 8 O k c l b a 1 m Figure 21 The thin and uniform plate 1 (see Fig. 21) of mass m can rotate about the horizontal axis O. The spring of stiﬀness k keeps it in the horizontal position. The damping coeﬃcient c reﬂects dissipation of energy of the system. Produce the formula for the natural frequency of the system.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 28 1.2 ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 1.2.1 Free vibration DEFINITION: It is said that a system performs free vibration if there are no external forces (forces that are explicitly dependent on time) acting on this system. In this section, according to the above deﬁnition, it is assumed that the resul- tant of all external forces f(t) is equal to zero. Hence, the mathematical model that is analyzed in this section takes form ¨x + 2ςωn ˙x + ω2 nx = 0 (1.28) The equation 1.28 is classiﬁed as linear homogeneous ordinary diﬀerential equation of second order. If one assume that the damping ratio ς is equal to zero, the equation 1.28 governs the free motion of the undamped system. ¨x + ω2 nx = 0 (1.29) Free vibration of an undamped system The general solution of the homogeneous equation 1.29 is a linear combination of its two particular linearly independent solutions. These solutions can be obtained by means of the following procedure. The particular solution can be predicted in the form 1.30. x = eλt (1.30) Introduction of the solution 1.30 into the equation 1.29 yields the characteristic equa- tion λ2 + ω2 n = 0 (1.31) This characteristic equation has two roots λ1 = +iωn and λ2 = −iωn (1.32) Hence, in this case, the independent particular solution are x1 = sin ωnt and x2 = cos ωnt (1.33) Their linear combination is the wanted general solution and approximates the free vibration of the undamped system. x = Cs sin ωnt + Cc cos ωnt (1.34) The two constants Cs and Cc should be chosen to fulﬁll the initial conditions which reﬂect the way the free vibrations were initiated. To get an unique solution it is nec- essary to specify the initial position and the initial velocity of the system considered. Hence, let us assume that at the instant t = 0 the system was at the position x0 and was forced to move with the initial velocity v0. Introduction of these initial conditions
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 29 into the equation 1.34 results in two algebraic equation that are linear with respect to the unknown constants Cs and Cc. Cc = x0 Csωn = v0 (1.35) According to 1.34, the particular solution that represents the free vibration of the system is x = v0 ωn sin ωnt + x0 cos ωnt = = C sin(ωnt + α) (1.36) where C = s (x0)2 + µ v0 ωn ¶2 ; α = arctan Ã x0 v0 ωn ! (1.37) For ωn = 1[1/s], x0 = 1[m] v0 = 1[m/s] and ς = 0 the free motion is shown in Fig. 22 The free motion, in the case considered is periodic. 40 -1.5 -1 -0.5 0 0.5 1 10 20 30 50 x[m] t[s] C Tn α xo vo Figure 22 DEFINITION: The shortest time after which parameters of motion repeat themselves is called period and the motion is called periodic motion. According to this deﬁnition, since the sine function has a period equal to 2π, we have sin(ωn(t + Tn) + α) = sin(ωnt + α + 2π) (1.38)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 30 Hence, the period of the undamped free vibrations is Tn = 2π ωn (1.39) Free vibration of a damped system If the damping ratio is not equal to zero, the equation of the free motion is ¨x + 2ςωn ˙x + ω2 nx = 0 (1.40) Introduction of the equation 1.30 into 1.40 yields the characteristic equation λ2 + 2ςωnλ + ω2 n = 0 (1.41) The characteristic equation has two roots λ1,2 = −2ςωn ± p (2ςωn)2 − 4ω2 n 2 = −ςωn ± ωn p ς2 − 1 (1.42) The particular solution depend on category of the above roots. Three cases are possible Case one - underdamped vibration If ς < 1, the characteristic equation has two complex conjugated roots and this case is often referred to as the underdamped vibration. λ1,2 = −ςωn ± iωn p 1 − ς2 = −ςωn ± iωd (1.43) where ωd = ωn p 1 − ς2 (1.44) The particular solutions are x1 = e−ςωnt sin ωdt and x2 = e−ςωnt cos ωdt (1.45) and their linear combination is x = e−ςωnt (Cs sin ωdt + Cc cos ωdt) (1.46) For the following initial conditions x |t=0= x0 ˙x |t=0= v0 (1.47) the two constants Cs and Cc are Cs = v0 + ςωnx0 ωd Cc = x0 (1.48)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 31 Introduction of the expressions 1.48 into 1.46 produces the free motion in the following form x = e−ςωnt (Cs sin ωdt + Cc cos ωdt) = Ce−ςωnt sin(ωdt + α) (1.49) where C = sµ v0 + ςωnx0 ωd ¶2 + (x0)2 ; α = arctan x0ωd v0 + ςωnx0 ; ωd = ωn p 1 − ς2 (1.50) For ωn = 1[1/s], x0 = 1[m] v0 = 1[m/s] and ς = .1 the free motion is shown in Fig. 23In this case the motion is not periodic but the time Td (see Fig. 23) between every -1.5 -1 -0.5 0 0.5 1 10 20 30 40 50 x[m] t[s] Td Td x(t)x(t+ Td t xo vo ) Figure 23 second zero-point is constant and it is called period of the dumped vibration. It is easy to see from the expression 1.49 that Td = 2π ωd (1.51) DEFINITION: Natural logarithm of ratio of two displacements x(t) and x(t + Td) that are one period apart is called logarithmic decrement of damping and will be denoted by δ.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 32 It will be shown that the logaritmic decrement is constant. Indeed δ = ln x (t) x (t + Td) = ln Ce−ςωnt sin(ωdt + α) Ce−ςωn(t+Td) sin(ωd(t + Td) + α) = = ln Ce−ςωnt sin(ωdt + α) Ce−ςωnte−ςωnTd sin(ωdt + 2π + α) = ςωnTd = 2πςωn ωd = 2πςωn ωn √ 1 − ς2 = = 2πς √ 1 − ς2 (1.52) This formula is frequently used for the experimental determination of the damping ratio ς. ς = δ p 4π2 + δ2 (1.53) The other parameter ωn that exists in the mathematical model 1.40 can be easily identiﬁed by measuring the period of the free motion Td. According to the formula 1.44 and 1.51 ωn = ωd √ 1 − ς2 = 2π Td √ 1 − ς2 (1.54) Case two - critically damped vibration If ς = 1, the characteristic equation has two real and equal one to each other roots and this case is often referred to as the critically damped vibration λ1,2 = −ςωn (1.55) The particular solutions are x1 = e−ςωnt and x2 = te−ςωnt (1.56) and their linear combination is x = Cse−ςωnt + Ccte−ςωnt (1.57) For the following initial conditions x |t=0= x0 ˙x |t=0= v0 (1.58) the two constants Cs and Cc are as follow Cs = x0 Cc = v0 + x0ωn (1.59) Introduction of the expressions 1.59 into 1.57 produces expression for the free motion in the following form x = e−ςωnt (x0 + t(v0 + x0ωn)) (1.60) For ωn = 1[1/s], x0 = 1[m] v0 = 1[m/s] and ς = 1. the free motion is shown in Fig. 24. The critical damping oﬀers for the system the possibly faster return to its equilibrium position.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 33 -1.5 -1 -0.5 0 0.5 1 10 20 30 40 50 x[m] t[s] vo xo Figure 24 Case three - overdamped vibration If ς > 1, the characteristic equation has two real roots and this case is often referred to as the overdamped vibration. λ1,2 = −ςωn ± ωn p ς2 − 1 = ωn(−ς ± p ς2 − 1) (1.61) The particular solutions are x1 = e−ωn(ς+ √ ς2−1)t and x2 = e−ωn(ς− √ ς2−1)t (1.62) and their linear combination is x = e−ωnt ³ Cseωn √ ς2−1)t + Cce−ωn √ ς2−1)t ´ (1.63) For the following initial conditions x |t=0= x0 ˙x |t=0= v0 (1.64) the two constants Cs and Cc are as follow Cs = + v0 ωn + x0(+ς + √ ς2 − 1) 2 √ ς2 − 1 Cc = − v0 ωn + x0(−ς + √ ς2 − 1) 2 √ ς2 − 1 (1.65) For ωn = 1[1/s], x0 = 1[m] v0 = 1[m/s] and ς = 5. the free motion is shown in Fig. 25
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 34 -1.5 -1 -0.5 0 0.5 1 10 20 30 40 50 x[m] t[s] vo xo Figure 25 1.2.2 Forced vibration In a general case motion of a vibrating system is due to both, the initial conditions and the exciting force. The mathematical model, according to the previous consideration, is the linear non-homogeneous diﬀerential equation of second order. ¨x + 2ςωn ˙x + ω2 nx = f(t) (1.66) where ωn = r k m ; 2ςωn = c m ; f(t) = Fex(t) m (1.67) The general solution of this mathematical model is a superposition of the general solution of the homogeneous equation xg and the particular solution of the non- homogeneous equation xp. x = xg + xp (1.68) The general solution of the homogeneous equation has been produced in the previous section and for the underdamped vibration it is xg = e−ςωnt (Cs sin ωdt + Cc cos ωdt) = Ce−ςωnt sin(ωdt + α) (1.69) To produce the particular solution of the non-homogeneous equation, let as assume that the excitation can be approximated by a harmonic function. Such a case is referred to as the harmonic excitation. f(t) = q sin ωt (1.70)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 35 In the above equation q represents the amplitude of the unit excitation and ω is the excitation frequency. Introduction of the expression 1.70 into equation 1.66 yields ¨x + 2ςωn ˙x + ω2 nx = q sin ωt (1.71) In this case it is easy to predict mode of the particular solution xp = As sin ωt + Ac cos ωt (1.72) where As and Ac are constant. The function 1.72 is the particular solution if and only if it fulﬁls the equation 1.71 for any instant of time. Therefore, implementing it in equation 1.71 one can get ¡ (ω2 n − ω2 )As − 2ςωnωAc ¢ sin ωt+ ¡ 2ςωnωAs + (ω2 n − ω2 )Ac ¢ cos ωt = q sin ωt (1.73) This relationship is fulﬁlled for any instant of time if (ω2 n − ω2 )As − 2ςωnωAc = q 2ςωnωAs + (ω2 n − ω2 )Ac = 0 (1.74) Solution of the above equations yields the expression for the constant As and Ac As = ¯ ¯ ¯ ¯ q −2ςωnω 0 (ω2 n − ω2 ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (ω2 n − ω2 ) −2ςωnω 2ςωnω (ω2 n − ω2 ) ¯ ¯ ¯ ¯ = (ω2 n − ω2 )q (ω2 n − ω2)2 + 4(ςωn)2ω2 Ac = ¯ ¯ ¯ ¯ (ω2 n − ω2 ) q 2ςωnω 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (ω2 n − ω2 ) −2ςωnω 2ςωnω (ω2 n − ω2 ) ¯ ¯ ¯ ¯ = −2(ςωn)ωq (ω2 n − ω2)2 + 4(ςωn)2ω2 (1.75) Introduction of the expressions 1.75 into the predicted solution 1.72 yields xp = As sin ωt + Ac cos ωt = A sin(ωt + ϕ) (1.76) where A = p A2 s + A2 c = q p (ω2 n − ω2)2 + 4(ςωn)2ω2 ϕ = arctan Ac As = − arctan 2(ςωn)ω ω2 n − ω2 (1.77) or A = q ω2 n q (1 − ( ω ωn )2)2 + 4ς2( ω ωn )2 ϕ = − arctan 2ς ω ωn 1 − ( ω ωn )2 (1.78) Introducing 1.69 and 1.76 into the 1.68 one can obtain the general solution of the equation of motion 1.71 in the following form x = Ce−ςωnt sin(ωdt + α) + A sin(ωt + ϕ) (1.79)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 36 The constants C and α should be chosen to fullﬁl the required initial conditions. For the following initial conditions x |t=0= x0 ˙x |t=0= v0 (1.80) one can get the following set of the algebraic equations for determination of the parameters C and α x0 = Co sin αo + A sin ϕ v0 = −Coςωn sin αo + Coωd cos αo + Aω cos ϕ (1.81) Introduction of the solution of the equations 1.81 (Co,αo) to the general solution, yields particular solution of the non-homogeneous equation that represents the forced vibration of the system considered. x = Coe−ςωnt sin(ωdt + αo) + A sin(ωt + ϕ) (1.82) This solution, for the following numerical data ς = 0.1, ωn = 1[1/s], ω = 2[1/s], Co = 1[m], αo = 1[rd], A = 0.165205[m], ϕ = 0.126835[rd] is shown in Fig. 26 (curve c).The solution 1.82 is assembled out of two terms. First term represents an -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 20 40 60 x[m] t[s] transient state of the forced vibration steady state of the forced vibration abc A Figure 26 oscillations with frequency equal to the natural frequency of the damped system ωd. Motion represented by this term, due to the existing damping, decays to zero (curve a in Fig. 1.82) and determines time of the transient state of the forced vibrations. Hence, after an usually short time, the transient state changes into the steady state represented by the second term in equation 1.82 (curve b in Fig. 1.82) x = A sin(ωt + ϕ) (1.83) This harmonic term has amplitude A determined by the formula 1.77. It does not depend on the initial conditions and is called amplitude of the forced vibration. Mo- tion approximated by the equation 1.83 is usually referred to as the system forced vibration.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 37 Both, the exciting force f(t) = q sin ωt (1.70) and the (steady state) forced vibration x = A sin(ωt + ϕ) (1.83) are harmonic. Therefore, they can be represented by means of two vectors ’rotating’ with the same angular velocity ω (see Fig. 27).One x A ω t ϕ q Asin( t+ )ϕω qsin( t )ω Figure 27 can see from the above interpretation that the angular displacement ϕ is the phase between the exciting force and the displacement it causes. Therefore ϕ is called phase of the forced vibration. Because the transient state, from engineering point of view play secondary role, in the following sections the steady state forced vibration will be considered only. Forced response due to rotating elements - force transmitted to foundation. x µ ω t mµ ω2 sinω t x M m mµ ω2 sinω t M k c R mµ ω2 a) b) Figure 28 One of many possible excitation of vibrations is excitation caused by inertia forces produced by moving elements. The possibly simplest case of vibration cased by this type of excitation is shown in Fig. 28. The rotor of an electrical motor rotates with the constant angular velocity ω. If µ represents the static imbalance of the rotor
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 38 and m is its mass, then the rotor produces the centrifugal force F = mµω2 (1.84) Its component along the vertical axis x is Fx = mµω2 sin ωt (1.85) The motor of mass M is supported by means of a beam of the stiﬀness k. The damping properties are approximated by the damping coeﬃcient c. Let us model vibration of the system. The physical model of the problem described is shown in Fig. 28b). Taking advantage of the earlier described method of formulation the mathematical model we have M ¨x = −kx − c ˙x + mµω2 sin ωt (1.86) Transformation of this equation into the standard form yields ¨x + 2ςωn ˙x + ω2 nx = q sin ωt (1.87) where ωn = r k M 2ςωn = c M q = mµω2 M (1.88) Hence, the steady state forced vibration are x = A sin(ωt + ϕ) (1.89) where according to 1.77 A = q ω2 n q (1 − ( ω ωn )2)2 + 4ς2( ω ωn )2 ϕ = − arctan 2ς ω ωn 1 − ( ω ωn )2 (1.90) or, taking into consideration 1.88 A = m M µ( ω ωn )2 q (1 − ( ω ωn )2)2 + 4ς2( ω ωn )2 ϕ = − arctan 2ς ω ωn 1 − ( ω ωn )2 (1.91) The ratio A m M µ , is called the magniﬁcation factor, Its magnitude and the phase ϕ as a function of the ratio ω ωn for diﬀerent damping factor ς is shown in Fig. 29.If the frequency of excitation changes from zero to the value equal to the natural frequency ωn, the amplitude of the forced vibration is growing. Its maximum depends on the damping ratio and appears for ω > ωn. The phenomenon at which amplitude of the forced vibration is maximum is called amplitude resonance. If the frequency of excitation tends towards inﬁnity, the amplitude of the forced vibration tends to m M µ. For ω = ωn, regardless the damping involved, phase of the forced vibration is equal to 90o . This phenomenon is called phase resonance. If the frequency of excitation
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 39 0 2 4 6 1 2 3 -180 -135 -90 -45 0 1 2 3 =0ζ =0.1ζ =0.25ζ =0.5ζ =1.0ζ =1.5ζ ϕ nω ω A µ m M 1 =0ζ =0.1ζ =0.25ζ =0.5ζ =1.0ζ =1.5ζ Figure 29
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 40 tends to inﬁnity, the phase tends to 180o . Hence the response of the system tends to be in the anti-phase with the excitation. The force transmitted to the foundation R, according to the physical model shown in Fig. 28b) is R(t) = kx + c ˙x = kA sin(ωt + ϕ) + cAω cos(ωt + ϕ) = A √ k2 + c2ω2 sin(ωt + ϕ + δ) (1.92) The amplitude of the reaction is |R| = A √ k2 + c2ω2 = AM p ω4 n + 4ς2ω2 nω2 = = mµω2 q 1 + 4ς2( ω ωn )2 q (1 − ( ω ωn )2)2 + 4ς2( ω ωn )2 The ampliﬁcation ratio |R| mµω2 of the reaction as a function of the ratio ω ωn is shown in Fig. 30.For the frequency of excitation ω < 1.4ωn the force transmitted to foundation 0 2 4 6 0 1 2 3 nω ω R µm 2 ω 1 1.4 =0ζ =0.1ζ =0.25ζ =0.5ζ =1.0ζ =1.5ζ Figure 30 is greater then the centrifugal force itself with its maximum close to frequency ωn. For ω > 1.4ωn this reaction is smaller then the excitation force and tends to zero when the frequency of excitation approaches inﬁnity. Forced response due to the kinematic excitation - vibration isolation The physical model of a system with the kinematic excitation is shown in Fig. 31b). Motion of the point B along the axis y causes vibration of the block M. This physical
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 41 L y M k cy x a v R G B a) b) Figure 31 model can be used to analyze vibration of a bus caused by the roughness of the surface of the road shown in Fig. 31a). The stiﬀness k of the spring and the damping coeﬃcient c represent the dynamic properties of the bus shock-absorbers. The block of mass M stands for the body of the bus. If the surface can be approximated by the sine-wave of the amplitude a and length L and the bus is travelling with the constant velocity v, the period of the harmonic excitation is T = L v (1.93) Hence, the frequency of excitation, according to 1.39 is ω = 2πv L (1.94) and the motion of the point B along the axis y can approximated as follows y = a sin ωt (1.95) The equation of motion of the bus is M ¨x = −kx − c ˙x + ky + c ˙y (1.96) Introduction of 1.95 yields M ¨x + c ˙x + kx = ka sin ωt + caω cos ωt or ¨x + 2ςωn ˙x + ω2 nx = ω2 na sin ωt + 2ςωnωa cos ωt = q sin(ωt + α) (1.97) where q = aω2 n r 1 + 4ς2( ω ωn )2 (1.98)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 42 Without any harm to the generality of the considerations one can neglect the phase α and adopt the mathematical model in the following form ¨x + 2ςωn ˙x + ω2 nx = q sin ωt (1.99) Motion of the block along axis x is governed by the equation 1.83 x = A sin(ωt + ϕ) where A = q ω2 n q (1 − ( ω ωn )2)2 + 4ς2( ω ωn )2 ϕ = − arctan 2ς ω ωn 1 − ( ω ωn )2 (1.100) Introduction of equation 1.98 gives A = a q 1 + 4ς2( ω ωn )2 q (1 − ( ω ωn )2)2 + 4ς2( ω ωn )2 ϕ = − arctan 2ς ω ωn 1 − ( ω ωn )2 (1.101) The magnifying factor A a and the phase ϕ as a function of ω ωn is shown in Fig. 32. For ω < 1.4ωn it is possible to arrange for the bus to have vibration smaller than the amplitude of the kinematic excitation The expression for the reaction force transmitted to the foundation is R = kx + c ˙x − ky − c ˙y = kA sin(ωt + ϕ) + cωA cos(ωt + ϕ) − ka sin ωt − cωa cos ωt = |R| sin(ωt + γ) (1.102) Problem of minimizing the reaction force R (e.g. 1.92) or the amplitude A (e.g.. 1.101) is called vibration isolation.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 43 -180 -135 -90 -45 0 1 2 3 ϕ nω ω A a 0 2 4 6 0 1 2 3 =0ζ =0.1ζ =0.25ζ =0.5ζ =1.0ζ =1.5ζ =0ζ =0.1ζ =0.25ζ =0.5ζ =1.0ζ =1.5ζ Figure 32
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 44 1.2.3 Problems Free vibrations Problem 9 1 2 H k c Figure 33 The carriage 1 of the lift shown in Fig. 33 operates between ﬂoors of a building. The distance between the highest and the lowest ﬂoor is H = 30m. The average mass of the carriage is m = 500kg. To attenuate the impact between the carriage and the basement in the case the rope 3 is broken, the shock absorber 2 is to be installed. Calculate the stiﬀness k and the damping coeﬃcient c of the shock-absorber which assure that the deceleration during the impact is smaller then 200m/s2 .
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 45 Solution In the worst case scenario, the lift is at the level H when the rope brakes. H k c x x mg Figure 34 Due to the gravity force the lift is falling down with the initial velocity equal to zero. Equation of motion of the lift is m¨x = mg (1.103) By double side by side integrating of the above equation one can get x = A + Bt + g 2 t2 (1.104) Introduction of the following initial conditions x |t=0= 0 ˙x |t=0= 0 yields A = 0 and B = 0 and results in the following equation of motion x = g 2 t2 (1.105) Hence, the time the lift reaches the shock-absorber is to = s 2H g (1.106) Since v = ˙x = gt (1.107)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 46 the velocity of the lift at the time of the impact with the shock-absorber is vo = ˙x |t=to = p 2Hg (1.108) To analyze the motion of the lift after impact let us introduce the inertial axis y in such a way that its origin coincides with the upper end of the shock-absorber at the instant of impact (see Fig. 35).Since at the instant of impact the spring k is H k c x y mg y Figure 35 uncompressed, the equation of motion after the lift has reached the shock-absorber is m¨y + c ˙y + ky = mg (1.109) or in the standardized form ¨y + 2ςωn ˙y + ω2 ny = g (1.110) where ωn = r k m ; 2ςωn = c m (1.111) It is easy to see that in the case considered the particular solution of the non- homogeneous equation is yp = g ω2 n (1.112) The best performance of the shock-absorber is expected if the damping is critical (ς = 1). In this case, there exists one double root and the general solution of the homogeneous equation is yg = C1e−ωnt + C2te−ωnt (1.113)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 47 Therefore the general solution of the non-homogeneous equation as the sum of yp and yg is y = C1e−ωnt + C2te−ωnt + g ω2 n (1.114) This equation has to fullﬁl the following initial conditions y |t=0= 0 ˙y |t=0= vo (1.115) Introduction of these initial conditions into the equation 1.113 yields C1 = − g ω2 n C2 = vo − g ωn (1.116) and results in the following equation of motion y = µ − g ω2 n ¶ e−ωnt + µ vo − g ωn ¶ te−ωnt + g ω2 n = g ω2 n ¡ 1 − e−ωnt ¢ + µ vo − g ωn ¶ te−ωnt = D ¡ 1 − e−ωnt ¢ + Ete−ωnt (1.117) where D = g ω2 n E = vo − g ωn (1.118) Double diﬀerentiation of the function 1.117 yields acceleration during the impact ¨y = ¡ −Dω2 n − 2Eωn ¢ e−ωnt + Eω2 nte−ωnt (1.119) By inspection of the function 1.118, one can see that the maximum of the deceleration occurs for time t = 0. Hence the maximum of deceleration is amax = ¨y |t=0= ¯ ¯−Dω2 n − 2Eωn ¯ ¯ (1.120) If vo > g ωn (1.121) both constants E and D are positive. Hence amax = Dω2 n + 2Eωn = g + 2voωn − 2g = 2voωn − g (1.122) This deceleration has to be smaller then the allowed deceleration aa = 200ms−2 . 2voωn − g < aa (1.123) It follows ωn < aa + g 2 √ 2Hg = 200 + 10 2 √ 2 · 30 · 10 = 4.28s−1 (1.124) Since ωn = q k m , the stiﬀness of the shock-absorber is k = ω2 nm = 4.282 · 500 = 9160N/m (1.125)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 48 and the damping coeﬃcient c = 2ςωnm = 2 · 1 · 4.28 · 500 = 4280Ns/m (1.126) Our computation can be accepted only if the inequality 1.121 is fullﬁl. Indeed vo = p 2Hg = √ 2 · 30 · 10 = 24.5 > g ωn = 10 4.28 = 2.4m/s (1.127) The displacement of the lift, its velocity and acceleration during the impact as a function of time is shown in Fig. 36
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 49 displacement time [s] y [m] 0 0.5 1 1.5 2 2.5 0 0.25 0.5 0.75 1 v [m/s] velocity time [s] -5 0 5 10 15 20 25 0.25 0.5 0.75 1 acceleration time [s] -200 -150 -100 -50 0 50 0.25 0.5 0.75 1 a [m/s ]2 Figure 36
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 50 Problem 10 The power winch W was mounted on the truss T as shown in Fig. 37a) To T R W k m c x a) b) Figure 37 analyze the vibrations of the power winch the installation was modelled by the one degree of freedom physical model shown in Fig. 38b). In this ﬁgure the equivalent mass, stiﬀness and damping coeﬃcient are denoted by m, k and c respectively. Origin of the axis x coincides with the centre of gravity of the weight m when the system rests in its equilibrium position. To identify the unknown parameters m, k, and c, the following experiment was carried out. The winch was loaded with the weight equal to M1 = 1000kg as shown in Fig. 38. Then the load was released allowing the installation to perform the vertical T R W Ml L Figure 38 oscillations in x direction. Record of those oscillations is presented in Fig. 39. Calculate the parameters m, k, and c.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 51 431 2 time [s] -0.003 -0.002 -0.001 0 0.001 0.002 0.003 x[m] Figure 39 Answer m = 7000kg; k = 3000000Nm−1 ; c = 15000Nsm−1
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 52 Problem 11 T R W k m c x a) b) Figure 40 The winch W shown in Fig. 40 is modelled as a system with one degree of freedom of mass m stiﬀness k and the damping coeﬃcient c. The winch is lifting the block of mass M with the constant velocity vo (see Fig. 41).Assuming that the rope T R W M Figure 41 R is not extendible produce expression for the tension in the rope R before and after the block will lose contact with the ﬂoor.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 53 Solution Tension in the rope R before the contact is lost In the ﬁrst stage of lifting the block M, it stays motionlessly at the ﬂoor whereas the lift itself is going down with respect to the inertial axis x with the constant velocity vo. The tension T in the rope R varies between 0 and Mg. 0 < T 0 Mg (1.128) If origin of the inertial axis x coincides with the gravity centre when the unloaded winch is at its equilibrium, the equation of motion of the winch is m¨x + c ˙x + kx = −T (1.129) In the above equation ¨x = 0 (the winch is moving with the constant velocity vo), ˙x = −vo and x = −vot. Hence −T = c(−vo) + k(−vot) (1.130) The equation 1.130 governs motion of the winch till the tension T will reach value Mg. Therefore the equation 1.130 allows the time of separation ts to be obtained. ts = Mg − cvo kvo (1.131) At the instant of separation the winch will be at the position determined by the following formula xs = −vots = − Mg − cvo k (1.132) If Mg < cvo then xs = ts = 0. If Mg > cvo T = cvo + kvot for 0 < t < ts (1.133) Tension in the rope R after the contact of the weight with the ﬂoor is lost Without any harm to the generality of the further consideration one may assume that the time corresponding to the instant of separation is equal to 0. For t > 0, the equation of motion of the winch and the block (see Fig. 42) are as following m¨x + c ˙x + kx = −T M ¨xb = T − Mg (1.134) Since the rope R is not extendible, the instantaneous length of the rope L is L = Lo − vot (1.135)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 54 k m c x T T Mg x xb L Figure 42 Where Lo stands for the initial length of the rope (the lenght the rope had at the instance t = 0). Taking into account that L = x − xb (1.136) we have xb = x − L = x − Lo + vot (1.137) Introduction of the equation 1.137 into 1.134 yields m¨x + c ˙x + kx = −T M ¨x = T − Mg (1.138) Elimination of the unknown tension force allows the equation of motion of the winch to be formulated (m + M)¨x + c ˙x + kx = −Mg (1.139) The standardized form is as following ¨x + 2ςωn ˙x + ω2 nx = q (1.140) where ωn = r k m + M ; 2ςωn = c m + M ; q = − Mg m + M (1.141) The particular solution of the non-homogeneous equation can be predicted as a con- stant magnitude A. Hence ω2 nA = q; A = q ω2 n (1.142)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 55 The general solution of the mathematical model 1.140 is x = e−ςωnt (Cs sin ωdt + Cc cos ωdt) + A (1.143) where ωd = ωn p 1 − ς2 (1.144) This solution has to fulﬁll the following initial conditions for t = 0 x = xs ˙x = −vo (1.145) Introduction of these initial conditions into the solution 1.143 yields the following expressions for the constants Cs and Cc Cs = −vo + ςωn(xs − A) ωd Cc = xs − A (1.146) Hence, x = e−ςωnt µ −vo + ςωn(xs − A) ωd sin ωdt + (xs − A) cos ωdt ¶ + A (1.147) The time history diagram of the above function is shown in Fig.43 The tension is x txs Td xmax A O Figure 43 determined by the equation 1.138 T = M ¨x + Mg (1.148)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 56 Double diﬀerentiation of the function 1.147 yields the wanted tension as a function of time T = Mg + Me−ςωnt ¡ Cs(ςωn)2 + 2Ccςωnωd − Csω2 d ¢ sin ωdt +Me−ςωnt ¡ Cc(ςωn)2 − 2Csςωnωd + Ccω2 d ¢ cos ωdt (1.149)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 57 Forced vibration Problem 12 E,I M m ω t l c µ B A Figure 44 The electric motor of mass M (see Fig. 44) is mounted on the massless beam of length l, the second moment of inertia of its cross-section I and Young modulus E. Shaft of the motor, of mass m, rotates with the constant angular velocity ω and its unbalance (distance between the axis of rotation and the shaft centre of gravity) is µ. The damping properties of the system are modelled by the linear damping of the damping coeﬃcient c. Produce expression for the amplitude of the forced vibration of the motor as well as the interaction forces transmitted to the foundation at the points A and B.
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 58 Solution E,I M m ω t l c µ x m ωµ 2 A B Figure 45 Application of the Newton’s approach to the system shown in Fig. 45 results in the following diﬀerential equations of motion. M ¨x = −kx − c ˙x + mµω2 sin ωt (1.150) where k stands for the stiﬀness of the beam EI. k = 48EI l3 (1.151) Its standardized form is ¨x + 2ςωn ˙x + ω2 nx = q sin ωt (1.152) where ωn = r k M 2ςωn = c m + M q = mµω2 M (1.153) The particular solution of the equation 1.152 x = A sin (ωt + ϕ) (1.154) where A = q ω2 n q (1 − ( ω ωn )2)2 + 4ς2( ω ωn )2 ϕ = − arctan 2ς ω ωn 1 − ( ω ωn )2 (1.155) represents the forced vibrations of the system. In the above formula A stands for the amplitude of the forced vibrations of the motor. The interaction force at the point A can be determined from equilibrium of forces acting on the beam at an arbitrarily chosen position x (see Fig 46).
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 59 E,I 0.5k l x x x0.5kx kx A D Figure 46 The force needed to displace the point D by x is equal to kx. Hence, the reaction at the point A is RA = −0.5kx = −0.5kA sin (ωt + ϕ) (1.156) B x x c c D xc x Figure 47 To move the point D (see Fig. 47) with the velocity ˙x the force c ˙x is required. Hence, from the equilibrium of the damper one can see that the reaction at the point B is RB = −c ˙x = −cωA sin (ωt + ϕ)
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 60 Problem 13 x z y z=x-y 1 2 3 4 5 Figure 48 Figure 48 presents a seismic transducer. Its base 2 is attached to the vibrating object 1. The seismic weight 3 of mass m is supported by the spring 4 of stiﬀness k and the damper 5 of the damping coeﬃcient c.This transducer records the displacement z = x − y (1.157) where y is the absolute displacement of the vibration object 1 and x is the absolute displacement of the seismic weight 3. Upon assuming that the object 1 performs a harmonic motion y = a sin ωt (1.158) derive the formula for the ampliﬁcation coeﬃcient κ of the amplitude of vibration of the object 1 of this transducer ( κ = amplitude of z amplitude of y ) as a function of the non-dimensional frequency ω ωn .
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 61 Solution The equation of motion of the system shown in Fig. 48 is m¨x + c ˙x + kx = c ˙y + ky (1.159) Its standardize form is ¨x + 2ςωn ˙x + ω2 nx = aqc cos ωt + aqs sin ωt (1.160) where ωn = r k m 2ςωn = c m qc = c m ω qs = k m (1.161) Simpliﬁcation of the right side of the above equation yields ¨x + 2ςωn ˙x + ω2 nx = aq sin (ωt + α) (1.162) where q = p q2 c + q2 s = ω2 n s 4ς2 µ ω ωn ¶2 + 1 α = arctan qc qs = arctan 2ς ω ωn (1.163) According to equation 1.76 (page 35) the particular solution of the equation 1.162 is xp = aA sin(ωt + α + ϕ) (1.164) where A = r 4ς2 ³ ω ωn ´2 + 1 sµ 1 − ³ ω ωn ´2 ¶2 + 4ς2 ³ ω ωn ´2 ϕ = − arctan 2ς ω ωn 1 − ³ ω ωn ´2 (1.165) Hence the record of the transducer is z = x − y = aA sin(ωt + α + ϕ) − a sin ωt = = aA cos (α + ϕ) sin ωt + aA sin (α + ϕ) cos ωt − a sin ωt = = (aA cos (α + ϕ) − a) sin ωt + aA sin (α + ϕ) cos ωt (1.166) The amplitude of this record is ampz = q (aA cos (α + ϕ) − a)2 + (aA sin (α + ϕ))2 = a p A2 + 1 − 2A cos(α + ϕ) (1.167) Therefore, the coeﬃcient of ampliﬁcation is κ = ampz ampy = p A2 + 1 − 2A cos(α + ϕ) (1.168) The diagram presented in Fig.49 shows this ampliﬁcation coeﬃcient κ as a function of the ratio ω ωn .If the coeﬃcient of ampliﬁcation κ is equal to one, the record of the
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 62 amplitude of vibration (ampz) is equal to the amplitude of vibration of the object (ampy = a). It almost happends, as one can see from the diagram 49, if the frequency ω of the recorded vibrations is twice greater than the natural frequency ωn of the transducer and the damping ratio ς is 0.25. 0 1 2 3 4 5 6 1 2 3 4 5 ω/ nω κ ζ=0.1 ζ=0.25 ζ=0.5 Figure 49
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 63 Problem 14 O A B α xa b c d k G C Figure 50 The physical model of a vibrating table is shown in Fig. 50. It can be con- sidered as a rigid body of the mass m and the moment of inertia about axis through its centre of gravity IG. It is supported with by means of the spring of the stiﬀness k and the damper of the damping coeﬃcient c. The motion of the lower end of the spring with respect to the absolute coordinate x can be approximated as follows x = X cos ωt where X stands for the amplitude of the oscillations of the point C and ω stands for the frequency of these oscillations. Produce: 1. the diﬀerential equation of motion of the vibrating table and present it in the standard form 2. the expression for the amplitude of the forced vibrations of the table caused by the motion of the point C 3. the expression for the interaction force at the point A 4. the expression for the driving force that has to be applied to the point C
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 64 Problem 15 Y 1 2 xG G L/2 L A B C c k L/2 aα Figure 51 Two uniform rods (1 and 2), each of length L and mass m, were joined together to form the pendulum whose physical model is depicted in Fig. 51. The pendulum performs small oscillations α about the axis through the point A. At the point B it is supported by a spring of stiﬀness k and a damper of damping coeﬃcient c. The point C of the damper is driven along the axis Y and its motion is approximated by the following function Y = A sin ωt Produce: 1. The expression for the position xG of the center of gravity G of the pendulum Answer: xG = 3 4 L − a 2. The expression for the moment of inertia of the pendulum about the axis through the point A. Answer: IA = 17 12 mL2 + 2maH2 − 3mLa 3. The diﬀerential equation of motion of the pendulum Answer: ¨α + 2ζωn ˙α + ω2 nα = q cos ωt
- ANALYSIS OF ONE-DEGREE-OF-FREEDOM SYSTEM 65 where: 2ζωn = ca2 IA ; ω2 n = 2mgxG+ka2 IA q = Aωca IA 4. The expression for the amplitude of the forced vibrations of the pendulum Answer: Aα = q ω2 n 1−( ω ωn ) 2 2 +4ζ2 ( ω ωn ) 2 5. The driving force that must be applied to the point C to assure the assumed motion . D = c(Aω cos ωt − aAα cos(ωt + ϕ)) ϕ = − arctan 2ζ ω ωn 1−( ω ωn ) 2
- Chapter 2 MECHANICAL VIBRATION OF MULTI-DEGREE-OF-FREEDOM LINEAR SYSTEMS Since in the nature massless or rigid elements do not exist, therefore each of the particle the real element is made of can moves independently. It follows that to determine its position with respect to the inertial space one has to introduce inﬁnite number of coordinates. Hence, according to the previously introduce deﬁnition, the number of degrees of freedom of each real element is equal to inﬁnity. But in many vibration problems, with acceptable accuracy, the real elements can be represented by a limited number of rigid elements connected to each other by means of massless elements representing the elastic and damping properties. This process is called discretization and the ﬁnal result of this process is called multi-degree-of-freedom system. In this chapter it will be assumed that forces produced by these massless mi ki ci mj xi kij cij xj y (t)i Figure 1 elements (springs and dampers) are linear functions of displacements and velocities respectively. 2.1 MODELLING 2.1.1 Physical model Fig. 1 shows part of a multi-degree-of-freedom system. Usually, to describe motion of such a system a set of local generalized coordinates is introduced. These coordinates (xi, xj, yi(t)) are motionless with respect to a global (inertial) system of coordinates
- MODELLING 67 (not shown in the Fig. 1). The coordinate yi(t) is not independent (is an explicit function of time) whereas the coordinates xi, and xj are independent and their number determines the number of degree of freedom of the system. Origin of each coordinate coincides with the centre of gravity of individual bodies when the whole system is at its equilibrium position. For this equilibrium position all the static forces acting on individual bodies produces the resultant force equal to zero. 2.1.2 Mathematical model It will be shown in this section that the equation of motion of the multi-degree of freedom linear system has the following form m¨x+c˙x+kx=F(t) where m - is the inertia matrix c - is the damping matrix k- is the stiﬀness matrix F - is the external excitation matrix x- is the displacement matrix There are many methods that allow the mathematical model to be formulated. In the following sections a few of them are presented. Newton-Euler method of formulation of the mathematical model To develop the equations of motion of the system described, one may utilize the Newton’s or Euler’s equations. Since in case considered the body of mass mi performs a plane motion hence the Newton’s equations may be used. mi ¨xi = F (2.1) If the system stays at its equilibrium position, as it was mention earlier, the resultant of all static forces is equal zero. Therefore, the force F must contains forces due to the displacement of the system from its equilibrium position only. To ﬁgure these forces out let us move the mass mi out of its equilibrium position by the displacement xi. The conﬁguration a) shown in the ﬁgure below is achieved.
- MODELLING 68 a) xi 6= 0 xj = 0 yi = 0 ˙xi = 0 ˙xj = 0 ˙yi = 0 b) xi = 0 xj 6= 0 yi = 0 ˙xi = 0 ˙xj = 0 ˙yi = 0 c) xi = 0 xj = 0 yi 6= 0 ˙xi = 0 ˙xj = 0 ˙yi = 0 d) xi = 0 xj = 0 yi = 0 ˙xi 6= 0 ˙xj = 0 ˙yi = 0 e) xi = 0 xj = 0 yi = 0 ˙xi = 0 ˙xj 6= 0 ˙yi = 0 f) xi = 0 xj = 0 yi = 0 ˙xi = 0 ˙xj = 0 ˙yi 6= 0 mi ki ci mj xi kij cij xj y (t)i xi kij i- x ki i- x mi ki ci mj xi kij cij xj y (t)i xj kij j+ x mi ki ci mj xi kij cij xj y (t)i yi ki i+ y mi ki ci mj xi kij cij xj y (t)i xi cij i- x ci i- x mi ki ci mj xi kij cij xj y (t)i xj cij j+ x mi ki ci mj xi kij cij xj y (t)i yj mi¨xi = −kixi − kijxi +kijxj +kiyi(t) −ci ˙xi − cij ˙xi +cij ˙xj +0 (2.2) Due to this displacement there are two forces kixi and kijxi acting on the considered mass mi. Both of them must be taken with sign ’-’ because the positive displacement xi causes forces opposite to the positive direction of axis xi. Similar consideration carried out for the displacements along the axis xj (conﬁguration b)) and axis yi ( conﬁguration c)) results in the term +kijxj. and +kiyi(t) respectively. Up to now it has been assumed that the velocities of the system along all coordinates are equal to zero and because of this the dampers do not produce any force. The last three conﬁgurations (d, e, and f) allow to take these forces into account. Due to motion of the system along the coordinate xi with velocity ˙xi two additional forces are created by the dampers ci and cij. they are −ci ˙xi and −cij ˙xi. Both of them are caused by positive velocity and have sense opposite to the positive sense of axis xi. Therefore they have to be taken with the sign ’-’. The forces caused by motion along the axis xj (conﬁguration e)) and axis yi ( conﬁguration f)) results in the term +cij ˙xj. and 0 respectively. Since the system is linear, one can add all this forces together to obtain mi¨xi = −kixi − kijxi + kijxj + kiyi(t) − ci ˙xi − cij ˙xi + cij ˙xj (2.3) After standardization we have the ﬁnal form of equation of motion of the mass mi. mi¨xi + (ci + cij) ˙xi − cij ˙xj + (ki + kij)xi − kijxj = kiyi(t) (2.4) To accomplished the mathematical model, one has to carry out similar consideration for each mass involved in the system. As a result of these consideration we are getting set of diﬀerential equation containing as many equations as the number of degree of freedom.
- MODELLING 69 Lagrange method of formulation of the mathematical model The same set of equation of motion one can get by utilization of the Lagrange’s equations d dt ( ∂ ∂ ˙qm T) − ∂ ∂qm T + ∂V ∂qm = Qm m = 1, 2, ....M (2.5) where T - is the system kinetic energy function V - stands for the potential energy function Qm - is the generalized force along the generalized coordinate qm The kinetic energy function of the system considered is equal to sum of the kinetic energy of the individual rigid bodies the system is made of. Hence T = IX i=1 ⎛ ⎝1 2 miv2 i + 1 2 £ ωix ωiy ωiz ¤ ⎡ ⎣ Iix 0 0 0 Iiy 0 0 0 Iiz ⎤ ⎦ ⎡ ⎣ ωix ωiy ωiz ⎤ ⎦ ⎞ ⎠ (2.6) where mi - mass of the rigid body vi - absolute velocity of the centre of gravity of the body ωix,ωiy,ωiz, - components of the absolute angular velocity of the body Iix, Iiy, Iiz - The principal moments of inertia of the body about axes through its centre of gravity Potential energy function V for the gravity force acting on the link i shown in Fig. 2 is Vi = migrGiZ (2.7) Z X YO rG i Gi i rGiZ Figure 2 Potential energy for the spring s of stiﬀness ks and uncompressed length ls (see Fig. 3) is Vs = 1 2 ks(|rA − rB| − ls)2 (2.8)
- MODELLING 70 Z X YO A B s rA rB Figure 3 Potential energy function for all conservative forces acting on the system is V = IX i=1 Vi + SX s=1 Vs (2.9) In a general case the damping forces should be classiﬁed as non-conservative ones and, as such, should be included in the generalized force Qm. It must be remembered that the Lagrange’s equations yield, in general case, a non-linear mathematical model. Therefore, before application of the developed in this chapter methods of analysis, the linearization process must be carried out. The following formula allows for any non- linear multi-variable function to be linearized in vicinity of the system equilibrium position qo 1, ...qo m, ...qo M f(q1, ...qm, ...qM , ˙q1, ... ˙qm, ... ˙qM ) = f(qo 1, ...qo m, ...qo M , 0, ...0, ...0)+ + PM m=1 ∂f ∂qm (qo 1, ...qo m, ...qo M , 0, ...0, ...0) ∆qm + PM m=1 ∂f ∂ ˙qm (qo 1, ...qo m, ...qo M , 0, ...0, ...0) ∆ ˙qm (2.10) In the case of the system shown in Fig. 1 the kinetic energy function is T = 1 2 mi ˙x2 i + 1 2 mj ˙x2 j + · · ·· (2.11) Dots in the above equation represents this part of the kinetic energy function that does not depend on the generalized coordinate xi. If the system takes an arbitral position that is shown in Fig. 4, elongation of the springs ki and kij are respectively ∆li = xi − yi ∆lij = xj − xi (2.12)
- MODELLING 71 mi ki ci mj xi kij cij xj y (t)i mi ki ci mj xi kij cij xj y (t)i xj xi y i Figure 4 Therefore, the potential energy function is V = 1 2 ki(xi − yi)2 + 1 2 kij(xj − xi)2 + · · ·· (2.13) Again, dots stands for this part of the potential energy function that does not depend on the generalized coordinate xi. It should be noted that the above potential energy function represents increment of the potential energy of the springs due to the dis- placement of the system from its equilibrium position. Therefore the above function does not include the potential energy due to the static deﬂection of the springs. It follows that the conservative forces due to the static deﬂections can not be produced from this potential energy function. They, together with the gravity forces, produce resultant equal to zero. Hence, if the potential energy due to the static deﬂections is not included in the function 2.13 the potential energy due to gravitation must not be included in the function 2.13 either. If the potential energy due to the static deﬂec- tions is included in the function 2.13 the potential energy due to gravitation must be included in the function 2.13 too. Generally, the force produced by the dampers is included in the generalized force Qm. But, very often, for convenience, a damping function (dissipation function) D is introduced into the Lagrange’s equation to produce the damping forces. The function D does not represent the dissipation energy but has such a property that its partial derivative produces the damping forces. The damping function is created by analogy to the creation of the potential energy function. The stiﬀness k is replaced by the damping coeﬃcient c and the generalized displacements are replaced by the generalized velocities. Hence, in the considered case, since the lower end of the damper is motionless, the damping function is D = 1 2 ci( ˙xi)2 + 1 2 cij( ˙xj − ˙xi)2 + · · ·· (2.14) The Lagrange’s equation with the damping function takes form d dt ( ∂ ∂ ˙qm T) − ∂ ∂qm T + ∂V ∂qm + ∂D ∂ ˙qm = Qm m = 1, 2, ....M (2.15)
- MODELLING 72 Introduction of the equations 2.11, 2.13 and 2.14 into equation 2.15 yields the equation of the motion of the mass mi. mi¨xi + (ci + cij) ˙xi − cij ˙xj + (ki + kij)xi − kijxj = kiyi(t) (2.16) The inﬂuence coeﬃcient method mj mi Fj xj xi xij Figure 5 Let us consider the ﬂexible structure shown in Fig. 5. Let us assume that the masses mi and mj can move along the coordinate xi and xj respectively. Let us apply to this system a static force Fj along the coordinate xj. Let xij be the displacement of the system along the coordinate xi caused by the force Fj. DEFINITION: The ratio δij = xij Fj (2.17) is called the inﬂuence coeﬃcient It can be easily proved (see Maxwell’s reciprocity theorem) that for any structure δij = δji (2.18) If one apply forces along all I generalized coordinates xi along which the system is allowed to move, the displacement along the i − th coordinate, according to the superposition principle, is. xi = IX j=1 δijFj i = 1, 2, ......I (2.19) These linear relationships can be written in the matrix form x = δF (2.20)
- MODELLING 73 The inverse transformation permits to produce forces that have to act on the system along the individual coordinates if the system is at an arbitrarily chosen position x. F = δ−1 x (2.21) The inverse matrix δ−1 is called stiﬀness matrix and will be denoted by k. k = δ−1 (2.22) Hence according to equation 2.21 is Fi = IX j=1 kijxj (2.23) If the system considered moves and its instantaneous position is determined by the vector x (x1, ....xj, ...xJ ) the force that acts on the particle mi is fi = −Fi = − IX j=1 kijxj (2.24) Hence, application of the third Newton’s law to the particle i yields the equation of its motion in the following form mi ¨xi + IX j=1 kijxj = 0 (2.25)
- MODELLING 74 2.1.3 Problems Problem 16 l 12 R Figure 6 The disk 1 of radius R, and mass m is attached to the massless beam 2 of radius r, length l and the Young modulus E as shown in Fig. 6 Develop equations of motion of this system.
- MODELLING 75 Solution. l 12 R y ϕy z Fd Md Figure 7 The motion of the disk shown in Fig. 7 is governed by Newton’s equations m¨y = Fd I ¨ϕy = Md (2.26) In the above mathematical model I = 1 4 mR2 - moment of inertia of the disk about axis x Fd, Md - forces acting on the disk due to its interaction with the beam The interaction forces Fd and Md can be expressed as a function of the dis- placements y and ϕy by means of the inﬂuence coeﬃcient method. l 2 z y y ϕy M Fs s Figure 8 If the beam is loaded with force Fs (see Fig. 8), the corresponding displace- ments y and ϕy are y = l3 3EJ Fs ϕy = l2 2EJ Fs (2.27) If the beam is loaded with force Ms (see Fig. 8), the corresponding displacements y and ϕy are y = l2 2EJ Ms, ϕy = l EJ Ms (2.28)
- MODELLING 76 Hence the total displacement along coordinates y and ϕy are y = l3 3EJ Fs + l2 2EJ Ms ϕy = l2 2EJ Fs + l EJ Ms (2.29) or in matrix form ∙ y ϕy ¸ = ∙ l3 3EJ l2 2EJ l2 2EJ l EJ ¸ ∙ Fs Ms ¸ (2.30) where J = π r4 4 (2.31) The inverse transformation yields the wanted forces as function of the displacements ∙ Fs Ms ¸ = ∙ l3 3EJ l2 2EJ l2 2EJ l EJ ¸−1 ∙ y ϕy ¸ = ∙ k11 k12 k21 k22 ¸ ∙ y ϕy ¸ (2.32) Since, according to the second Newton’s law ∙ Fd Md ¸ = − ∙ Fs Ms ¸ (2.33) the equation of motion takes the following form ∙ m 0 0 I ¸ ∙ ¨y ¨ϕy ¸ = − ∙ k11 k12 k21 k22 ¸ ∙ y ϕy ¸ (2.34) Hence, the ﬁnal mathematical model of the system considered is m¨x + kx = 0 (2.35) where m = ∙ m 0 0 I ¸ ; k = ∙ k11 k12 k21 k22 ¸ ; x = ∙ y ϕy ¸ (2.36)
- MODELLING 77 Problem 17 k G k1 l1 2 l2 Figure 9 A rigid beam of mass m and the moments of inertia I about axis through its centre of gravity G is supported by massless springs k1, and as shown in Fig. 9. Produce equations of motion of the system.
- MODELLING 78 Solution. k G y 1 l1 l2 k2 y ϕ y =y+ϕ l22ϕ l11 O F M y =y- Figure 10 The system has two degree of freedom. Let us then introduce the two coordi- nates y and ϕ as shown in Fig. 10. The force F and the moment M that act on the beam due to its motion along coordinates y and ϕ are F = −y1k1 − y2k2 = −(y − ϕl1)k1 − (y + ϕl2)k2 = −[(k1 + k2)y + (k2l2 − k1l1)ϕ] M = +y1k1l1 − y2k2l2 = +(y − ϕl1)k1l1 − (y + ϕl2)k2l2 = −[(k2l2 − k1l1)y + (k1l2 1 + yk2l2 2)ϕ (2.37) Hence, the generalized Newton’s equations yield m¨y = F = −[(k11 + k2)y + (k2l2 − k1l1)ϕ] I ¨ϕ = M = −[(k2l2 − k1l1)y + (k1l2 1 + yk2l2 2)ϕ (2.38) The matrix form of the system equations of motion is m¨x + kx = 0 (2.39) where m = ∙ m 0 0 I ¸ ; k = ∙ k1 + k2 k2l2 − k1l1 k2l2 − k1l1 k1l2 1 + k2l2 2 ¸ ; x = ∙ y ϕ ¸ (2.40)
- MODELLING 79 Problem 18 1 2 k k Rr3 4 A Figure 11 The link 1 of a mass m1, shown in Fig. 11), can move along the horizontal slide and is supported by two springs 3 each of stiﬀness k. The ball 2 of mass m2 and a radius r and the massless rod 4 form a rigid body. This body is hinged to the link 1 at the point A. All motion is in the vertical plane. Use Lagrange’s approach to derive equations of small vibrations of the system about its equilibrium position. I = 2 5 m2r2 – moment of inertia of the ball about axis through its centre of gravity.
- MODELLING 80 Solution 1 2 k k Rr 3 x y x o G rG ϕ Figure 12 The system has two degree of freedom and the two generalized coordinates x and ϕ are shown in Fig. 12. The kinetic energy of the system T is equal to the sum of the kinetic energy of the link 1 T1 and the link 2 T2. T = T1 + T2 = 1 2 m1 ˙x2 + 1 2 m2v2 G + 1 2 I ˙ϕ2 (2.41) The absolute velocity of the centre of gravity of the ball vG can be obtained by diﬀerentiation of its absolute position vector. According to Fig .12, this position vector is rG = i(x + R sin ϕ) + j( − R cos ϕ) (2.42) Hence vG = ˙rG = i( ˙x + R ˙ϕ cos ϕ) + j(R ˙ϕ sin ϕ) (2.43) The required squared magnitude of this velocity is v2 G = ( ˙x + R ˙ϕ cos ϕ)2 + (R ˙ϕ sin ϕ)2 = ˙x2 + 2 ˙xR ˙ϕ cos ϕ + R2 ˙ϕ2 (2.44) Introduction of Eq. 2.44 into Eq. 2.41 yields the kinetic energy function of the system as a function of the generalized coordinates x and ϕ. T = 1 2 m1 ˙x2 + 1 2 m2( ˙x2 + 2 ˙xR ˙ϕ cos ϕ + R2 ˙ϕ2 ) + 1 2 I ˙ϕ2 = 1 2 (m1 + m2) ˙x2 + m2R ˙x ˙ϕ cos ϕ + 1 2 (m2R2 + I) ˙ϕ2 (2.45) The potential energy function is due the energy stored in the springs and the energy due to gravitation. V = 2 1 2 kx2 − m2gR cos ϕ (2.46)
- MODELLING 81 In the case considered, the Lagrange’s equations can be adopted in the following form d dt µ ∂T ∂ ˙x ¶ − ∂T ∂x + ∂V ∂x = 0 d dt µ ∂T ∂ ˙ϕ ¶ − ∂T ∂ϕ + ∂V ∂ϕ = 0 (2.47) The individual terms that appeare in the above equation are d dt µ ∂T ∂ ˙x ¶ = d dt ((m1 + m2) ˙x + m2R ˙ϕ cos ϕ) = = (m1 + m2)¨x + m2R¨ϕ cos ϕ − m2R ˙ϕ2 sin ϕ (2.48) ∂T ∂x = 0 (2.49) ∂V ∂x = 2kx (2.50) d dt µ ∂T ∂ ˙ϕ ¶ = d dt ¡ m2R ˙x cos ϕ + (m2R2 + I) ˙ϕ ¢ = = (m2R2 + I)¨ϕ + m2R¨x cos ϕ − m2R ˙x ˙ϕ sin ϕ (2.51) ∂T ∂ϕ = −m2R ˙x ˙ϕ sin ϕ (2.52) ∂V ∂ϕ = m2gR sin ϕ (2.53) Hence, according to Eq. 2.47, we have the following equations of motion (m1 + m2)¨x + m2R¨ϕ cos ϕ − m2R ˙ϕ2 sin ϕ + 2kx = 0 (m2R2 + I)¨ϕ + m2R¨x cos ϕ + m2gR sin ϕ = 0 (2.54) For small magnitudes of x and ϕ, sinϕ ∼= ϕ, cos ϕ ∼= 1, ˙ϕ2 ∼= 0. Taking this into account the linearized equations of motion are (m1 + m2)¨x + m2R¨ϕ + 2kx = 0 (m2R2 + I)¨ϕ + m2R¨x + m2gRϕ = 0 (2.55) Their matrix form is m¨x + kx = 0 (2.56) where m = ∙ m1 + m2 m2R m2R m2R2 + I ¸ , k = ∙ 2k 0 0 m2gR ¸ , x = ∙ x ϕ ¸ (2.57)
- MODELLING 82 Problem 19 k k k k q 1 q 2 l l A1 A2 Figure 13 Two identical and uniform rods shown in Fig. 13, each of mass m and length l, are joined together to form an inverse double pendulum. The pendulum is supported by four springs, all of stiﬀness k, in such way that its vertical position (q1 = 0 and q2 = 0) is its stable equilibrium position. Produce equation of small vibrations of the pendulum about this equilibrium position.
- MODELLING 83 Problem 20 l 1 2 EI 4 3 4 R l3 GJo Figure 14 The disk 1 of mass m1 and radius R shown in Fig. 14, is fasten to the massless and ﬂexible shaft 3. The left hand end of the massless and ﬂexible beam 4 is rigidly attached to the disk 1. At its right hand side the particle 2 of m2 is placed. Derive equations for analysis of small vibrations of the system.
- MODELLING 84 Problem 21 k R I2 R k I1 I3 J2l 2 G2J1l 1 G1 Figure 15 A belt gear was modelled as shown in Fig. 15. The shafts are assumed to be massless and their length the second moment of inertia and the shear modulus is denoted by l, J, and G respectively. The disks have moments of inertia I1, I2, and I3. The belt is modelled as the spring of a stiﬀness k. Derive the diﬀerential equations for the torsional vibrations of the system.
- MODELLING 85 Problem 22 D I1 I2 I3 J2l 2 G2J1l 1 G1 1 D2 Figure 16 In Fig. 16 the physical model of a gear box is presented. Derive equations for the torsional vibrations of the gear box. The shafts the gears are mounted on are massless.
- MODELLING 86 Problem 23 B O Y X C α ll l l l l A 12 34 56 7 Figure 17 Fig. 17 shows a mechanical system. Link 1 of the system is motionless with respect to the inertial system of coordinates XY . The links 2 and 3 are hinged to the link 1 at the point O. The links 4 and 5 join the links 2 and 3 with the collar 6. The spring 7 has a stiﬀness k and its uncompressed length is equal to 2l. The system has one degree of freedom and its position may be determined by one generalized coordinate α. The links 4 5 and 6 are assumed to be massless. The links 2 and 3 can be treated as thin and uniform bars each of length 2l and mass m. Derive equations of the small vibration of the system about its equilibrium position.
- MODELLING 87 Problem 24 m m m T l 4 l 4 l 4 l 4 Figure 18 Three beads, each of mass m are attached to the massless string shown in Fig. 18. The string has length l and is loaded with the tensile force T. Derive equation of motion of the beads
- MODELLING 88 Problem 25 Rq 1 q 2 l m Figure 19 On the massless string of length l the ball of mass m and radius R is suspended (see Fig. 19). Derive equation of motion of the system.
- MODELLING 89 Problem 26 k c m ks1 s2k 2I1I1I 2I ii R Figure 20 Fig. 20 presents the physical model of a winch. The shafts of the torsional stiﬀness ks1 and ks2 as well as the gear of ratio i are massless. To the right hand end of the shaft ks2 the rotor of the moment of inertia I2 is attached. The left hand end of the shaft ks1 is connected to the drum of the moment of inertia I1. The rope is modelled as a massless spring of the stiﬀness k. At its end the block of mass m is fastened. The damper of the damping coeﬃcient c represents the damping properties of the system. Produce the diﬀerential equation of motion of the system.
- MODELLING 90 Solution k c m ks1 s2k 2I1I2I ii R x α1 α2 ϕ2 ϕ1 F F r2 r1 Figure 21 In Fig. 21 x, ϕ1and ϕ2 are the independant coordinates. Since the gear of the gear ratio as well as the shafts of stiﬀness ks1 and ks2 are massless the coordinates that specify the position of the of the gear α1 and α2 are not independent. They are a function of the independent coordinates. Let i = r1 r2 = α2 α1 (2.58) Application of the Newton’s law to the individual bodies yields equations of motion in the following form. m¨x = −kx + Rkϕ1 (2.59) I1 ¨ϕ1 = +kRx − kR2 ϕ1 − cR2 ˙ϕ1 − ks1ϕ1 + ks1α1 (2.60) 0¨α1 = −ks1α1 + ks1ϕ1 + Fr1 (2.61) 0¨α2 = −ks2α2 + ks2ϕ2 − Fr2 (2.62) I2 ¨ϕ2 = −ks2ϕ2 + ks2α2 (2.63) Introducing 2.58 into the equations 2.61 and 2.62 one can obtain 0 = −ks1α1 + ks1ϕ1 + iFr2 (2.64) 0 = −ks2iα1 + ks2ϕ2 − Fr2 (2.65) Solving the above equations for α1 we have α1 = ks1ϕ1 + iks2ϕ2 ks1 + ks2i2 (2.66)
- MODELLING 91 Hence, according to 2.58 α2 = iα1 = iks1ϕ1 + i2 ks2ϕ2 ks1 + ks2i2 (2.67) Introducing 2.66 and 2.67 into 2.60 and 2.63 one can get the equations of motion in the following form m¨x = −kx + Rkϕ1 I1 ¨ϕ1 = +kRx − kR2 ϕ1 − cR2 ˙ϕ1 − ks1ϕ1 + ks1 ks1ϕ1 + iks2ϕ2 ks1 + ks2i2 (2.68) ¨ϕ2 = −ks2ϕ2 + ks2 iks1ϕ1 + i2 ks2ϕ2 ks1 + ks2i2 After standardization we have m¨z + c˙z + kz = 0 where m = ⎡ ⎣ m 0 0 0 I1 0 0 0 I2i2 ⎤ ⎦ ; c = ⎡ ⎣ 0 0 0 0 cR2 0 0 0 0 ⎤ ⎦ ; (2.69) k = ⎡ ⎢ ⎣ k −kR 0 −kR kR2 + ks1ks2i2 ks1+ks2i2 − ks1ks2i2 ks1+ks2i2 0 − ks1ks2i2 ks1+ks2i2 ks1ks2i2 ks1+ks2i2 ⎤ ⎥ ⎦ ; z = ⎡ ⎣ x ϕ1 ϕ2 i ⎤ ⎦
- MODELLING 92 Problem 27 Z X x z G C R 4R 3π O α Figure 22 The semi-cylinder of mass m and radius R shown in Fig. 50 is free to roll over the horizontal plane XY without slipping. The instantaneous angular position of this semi-cylinder is determined by the angular displacement α. Produce 1. the equation of small oscillations of the semi-cylinder (take advantage of the Lagrange’s equations) Answer:¡ IG + mR2 ¡ 1 + 16 9π2 − 8 3π ¢¢ ¨α + mgR(1 − 4 3π )α = 0 where IG = 1 2 mR2 − m ¡4 3 R π ¢2 2. the expression for period of these oscillations. Answer: T = 2π mgR(1− 4 3π ) (IG+mR2 (1+ 16 9π2 − 8 3π ))
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 93 Problem 28 I1 I2 J3l3 G3 J1l1 G1 J2l2 G2 J4l4 G4 Figure 23 The two disks of moments of inertia I1 and I2 are join together by means of the massless shafts as is shown in Fig. 51. The dynamic properties of the shafts are determined by their lenghts l, the second moments of area J and the shear modulus G. Produce the diﬀerential equations of motion. 2.2 ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM The analysis carried out in the previous section leads to conclusion that the mathe- matical model of the linear multi-degree-of -freedom system is as follows m¨x + c˙x + kx = F(t) (2.70) where m - matrix of inertia c - matrix of damping k - matrix of stiﬀness F(t)- vector of the external excitation x- vector of the generalized coordinates 2.2.1 General case In the general case of the multi-degree-of-freedom system the matrices c and k do not necessary have to be symmetrical. Such a situation takes place, for example, if the mechanical structure interacts with ﬂuid or air (oil bearings, ﬂatter of plane wings etc.). Since the equation 2.70 is linear, its general solution is always equal to the sum of the general solution of the homogeneous equation xg and the particular solution of the non-homogeneous equation xp. x = xg + xp (2.71) The homogeneous equation m¨x + c˙x + kx = 0 (2.72)
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 94 corresponds to the case when the excitation F(t) is not present. Therefore, its gen- eral solution represents the free (natural) vibrations of the system. The particular solution of the non-homogeneous equation 2.70 represents the vibrations caused by the excitation force F(t). It is often refered to as the forced vibrations. Free vibrations - natural frequencies- stability of the equilibrium position To analyze the free vibrations let us transfer the homogeneous equation 2.72 to so called state-space coordinates. Let y = ˙x (2.73) be the vector of the generalized velocities. Introduction of Eq. 2.73 into Eq. 2.72 yields the following set of the diﬀerential equations of ﬁrst order. ˙x = y ˙y = −m−1 kx − m−1 cy (2.74) The above equations can be rewritten as follows ˙z = Az (2.75) where z = ∙ x y ¸ , A = ∙ 0 1 −m−1 k −m−1 c ¸ (2.76) Solution of the above equation can be predicted in the form 2.77. z = z0ert (2.77) Introduction of Eq. 2.77 into Eq. 2.75 results in a set of the homogeneous algebraic equations which are linear with respect to the vector z0. [A − 1r] z0 = 0 (2.78) The equations 2.78 have non-zero solution if and only if the characteristic determinant is equal to 0. |[A − 1r]| = 0 (2.79) The process of searching for a solution of the equation 2.79 is called eigenvalue problem and the process of searching for the corresponding vector z0 is called eigenvector problem. Both of them can be easily solved by means of the commercially available computer programs. The roots rn are usually complex and conjugated. rn = hn ± iωn n = 1.....N (2.80) Their number N is equal to the number of degree of freedom of the system considered. The particular solutions corresponding to the complex roots 2.80 are zn1 = ehnt (Re(z0n) cos ωnt − Im(z0n) sin ωnt) zn2 = ehnt (Re(z0n) sin ωnt + Im(z0n) cos ωnt) n = 1.....N (2.81)
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 95 In the above expressions Re(z0n ) and Im(z0n ) stand for the real and imaginary part of the complex and conjugated eigenvector z0n associated with the nth root of the set 2.80 respectively. The particular solutions 2.81 allow to formulate the general solution that approximates the system free vibrations.. z = [z11, z12,z21, z22,z31, z32,.....zn1, zn2,.........zN1, zN2] C (2.82) As one can see from the formulae 2.81, the imaginary parts of roots rn represent the natural frequencies of the system and their real parts represent rate of decay of the free vibrations. The system with N degree of freedom possesses N natural frequencies. The equation 2.82 indicates that the free motion of a multi- degree-of-freedom system is a linear combination of the solutions 2.81. A graphical interpretation of the solutions 2.81 is given in Fig. 24 for the positive and negative magnitude of hn.The problem of searching for the vector of the -20 -15 -10 -5 0 5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 t z h>0 /π ω nnT = 2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 t z /π ω nnT = 2 h<0 Figure 24 constant magnitudes C is called initial problem. In the general case, this problem is diﬃcult and goes beyond the scope of this lectures. The roots 2.80 allow the stability of the system equilibrium position to be determined. If all roots rn of the equation 2.80 have negative real parts then the equilibrium position of the system considered is stable. If at least one root of the equation 2.80 has positive real part then the equilibrium position of the system considered is unstable. Forced vibrations - transfer functions The response to the external excitation F(t) of a multi-degree-of-freedom system is determined by the particular solution of the mathematical model 2.70. m¨x + c˙x + kx = F(t) (2.83) Let us assume that the excitation force F(t) is a sum of K addends. For the further analysis let us assume that each of them has the following form Fk = Fk o cos(ωt + ϕk o) (2.84) To facilitate the process of looking for the particular solution of equation 2.83, let us introduce the complex excitation force by adding to the expression 2.84 the imaginary part. fk = Fk o cos(ωt + ϕk o) + iFk o sin(ωt + ϕk o) (2.85)
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 96 The relationship between the complex excitation fk and the real excitation is shown in Fig. 25. According to Euler’s formula the complex excitation may be rewritten as ω k f t 0 k F ϕ0 k Re Im 0 k F cos( )ω t+ϕ0 k k F ω t ϕ0 k ω t 0 k F Figure 25 follows fk = Fk o ei(ωt+ϕk o ) = Fk o eiϕk o eiωt = fk o eiωt (2.86) Here, fk o is a complex number that depends on the amplitude and phase of the external excitation. Introduction of Eq. 2.86 into Eq. 2.83 yields m¨x + c˙x + kx = foeiωt (2.87) Now, the particular solution of Eq. 2.87 can be predicted in the complex form 2.88 xc = aeiωt (2.88)
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 97 Introduction of Eq. 2.88 into Eq. 2.87 produces set of the algebraic equations which are linear with respect to the unknown vector a. ¡ −ω2 m + iωc + k ¢ a = fo (2.89) Its solution is a = ¡ −ω2 m + iωc + k ¢− 1 fo (2.90) Therefore, according to Eq. 2.88, the response of the system xc due to the complex force f is xc = (Re(a) + i Im(a))(cos ωt + i sin ωt) (2.91) Response of the system x due to the real excitation F is represented by the real part of the solution 2.91. x = Re(a) cos ωt − Im(a) sin ωt (2.92) Motion of the system considered along the coordinate xk , according to 2.92 is xk = xk o cos(ωt + βk ) (2.93) where xk o = p Re(ak)2 + Im(ak)2 βk = arc tan Im(ak ) Re(ak) (2.94) It is easy to see from 2.91 that the amplitude of the forced vibration xk o is equal to the absolute value of the complex amplitude ak , and its phase βk is equal to the phase between the complex amplitudes ak and the vector eiωt . This ﬁndings are presented in Fig. 26. The complex matrix ¡ −ω2 m + iωc + k ¢− 1 (2.95) will be denoted by R(iω) and it is called matrix of transfer functions. It transfers, according to 2.90, the vector of the complex excitation foeiωt into the vector of the complex displacement xc = aeiωt . xc = aeiωt = R(iω)foeiωt (2.96) It is easy to see that the element Rpq(iω) of the matrix of transfer functions represents the complex displacement (amplitude and phase) of the system along the coordinates xp caused by the unit excitation 1eiωt along the coordinate xq. Example of three elements of a matrix of the transform functions are presented in Fig. 27. The ﬁrst two diagrams present the real and the imaginary parts of the complex transform functions whereas the last two present its absolute value (amplitude) and phase.
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 98 ω k f t 0 k F ϕ0 k Re Im 0 k F cos( )ω t+ϕ0 k k F ω t ϕ0 k ω t 0 k F β0 k β0 k 0 k x 0 k x 0 k x cos( )ω t+β0 k k x k a Figure 26
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 99 -0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 500 1000 1500 frequency rad/s transfer function m/N (imaginary part) R(1,1) R(1,2) R(1,3) -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 500 1000 1500 frequency rad/s transfer functions m/N (real parts) R(1,1) R(1,2) R(1,3) 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0 500 1000 1500 frequency rad/s transfer function m/N (modulus) R(1,1) R(1,2) R(1,3) -4 -2 0 2 4 500 1000 1500 frequency rad/s transfer functions m/N (phase) R(1,1) R(1,2) R(1,3) Figure 27
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 100 Experimental determination of the transfer functions F(t) x(t) t t Fourier transformation F x(p q p q ( ) ) iω iω ω ω R i pq (iω ) ___ x(p )iω Fq ( )iω ω Fq xq xp x p Figure 28 In order to produce the transfer function between the coordinate xp and the coor- dinate xq (see Fig. 28) let us apply force Fq(t) along the coordinate xq and record it simultaneously with the system response xp(t) along the coordinate xp. Fourier transformation applied to these functions Fq(iω) = Z +∞ −∞ e−iωt Fq(t)dt xp(iω) = Z +∞ −∞ e−iωt xp(t)dt (2.97) yields the Fourier transforms in the frequency domain xp(iω) and Fq(iω). The am- plitude of the complex functions xp(iω) and Fq(iω) |xp(iω)| = q Re(xp(iω))2 + Im(xp(iω))2 |Fq(iω)| = q Re(Fq(iω))2 + Im(Fq(iω))2 (2.98)
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 101 represents the amplitude of displacement and force respectively as a function of the frequency ω. The corresponding phases are determined by the following formulae. ϕxp = arctan Im(xp(iω)) Re(xp(iω)) ϕFq = arctan Im(Fq(iω)) Re(Fq(iω)) (2.99) These Fourier transforms allow the transfer function Rpq(iω) to be computed. Rpq(iω) = xp(iω) Fq(iω) (2.100) The above formula determines response of the system along coordinate xp caused by the harmonic excitation Fq along the coordinate xq. xp(iω) = Rpq(iω)Fq(iω) (2.101) Since the system considered is by assumption linear, the response along the coordinate xp caused by set of forces acting along coordinates N coordinates xq, according to the superposition principle, is xp(iω) = q=N X q=1 Rpq(iω)Fq(iω) q = 1....N (2.102) Application of the above described experimental procedure to all coordinates involved in the modelling (p = 1....N) allows to formulate the matrix of the transfer functions Rpq(iω).The relationship above can be rewritten in the following matrix form x(iω) = Rpq(iω)F(iω) p = 1....N, q = 1....N (2.103)
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 102 2.2.2 Modal analysis - case of small damping In the following analysis it will be assumed that the matrices m, c and k are square and symmetrical. Size of these matrices is N × N where N is the number of the system degree of freedom. If the vector of the external excitation F(t) is equal to zero, it is said that the system performs free vibrations. According to the above deﬁnition the free vibrations are governed by the homogeneous set of equations m¨x + c˙x + kx = 0 (2.104) Free vibration of the undamped system - eigenvalue and eigenvector prob- lem If the damping is neglected the equation of the free vibrations is m¨x + kx = 0 (2.105) It is easy to see that x = X cos ωt (2.106) is a particular solution of the equation 2.105. Indeed, introduction of Eq. 2.106 into 2.105 yields (−ω2 m + k)X cos ωt = 0 (2.107) and the diﬀerential equation 2.105 is fulﬁlled for any instant of time if the following set of the homogeneous algebraic equations is fulﬁlled. (−ω2 m + k)X = 0 (2.108) In turn, the above set of equations has the non-zero solutions if and only if its char- acteristic determinant is equal to zero ¯ ¯−ω2 m + k ¯ ¯ = 0 (2.109) The above characteristic equation, for any physical system, has N positive roots with respect to the parameter ω2 . Hence, the parameter ω can take any of the following values ±ω1, ±ω2, ±ω3, ..... ± ωn, .... ± ωN (2.110) As one can see from Eq. 2.106, these parameters have the physical meaning only for positive values. They represent frequencies of the system free vibrations. They are called natural frequencies. The number of diﬀerent natural frequencies is therefore equal to the number of degree of freedom. For each of the possible natural frequencies ωn the system of equations 2.108 becomes linearly dependent and therefore has inﬁnite number of solution Xn. Its follows that if Xn is a solution of Eq. 2.108, the vector CnXn (2.111) where Cn is arbitrarily chosen constant, is solution of the Eq. 2.108 too. The vec- tor Xn represents so called natural mode of vibration associated with the natural
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 103 frequency ωn. It determines the shape that the system must possess to oscillate harmonically with the frequency ωn. For example, if a beam with four concentrated masses is considered (see Fig. 29) the vector Xn contains four numbers Xn = [X1n, X2n, X3n, X4n]T (2.112) If the system is deﬂected according to the this vector and allowed to move with the Tn X1n X2n X3n X4n x 3nx = X cos t3n ωn t Figure 29 initial velocity equal to zero, it will oscillate with the frequency ωn. There are four such a natural modes and four corresponding natural frequencies for this system. The problem of the determination of the natural frequencies is called eigen- value problem and searching for the corresponding natural modes is called eigenvector problem. Therefore the natural frequencies are very often referred to as eigenvalue and the natural modes as eigenvectors. Now, one can say that the process of determination of the particular solution xn= Xn cos ωnt (2.113) of the equation 2.105 has been accomplished. There are N such particular solutions. In similar manner one can prove that xn= Xn sin ωnt (2.114) is a particular solution too. Since the solutions ?? and 2.114 are linearly independent, their linear combination forms the general solution of the equation 2.105 xn = NX n=1 (SnXn sin ωnt + CnXn cos ωnt) (2.115) The 2N constants Sn and Cn should be chosen to satisfy the 2N initial conditions.
- ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEM 104 Properties of the natural modes. Each eigenvector has to fulﬁll the Eq. 2.108. Hence, −ω2 nmXn+kXn = 0 −ω2 mmXm+kXm = 0 (2.116) Primultiplying the ﬁrst equation by XT m and the second equation by XT n one can get −ω2 nXT mmXn+XT mkXn = 0 −ω2 mXT n mXm+XT n kXm = 0 (2.117) Since matrices m and k are symmetrical XT mkXn = XT n kXm and XT mmXn = XT n mXm (2.118) Now, primultiplying the ﬁrst equation of set 2.117 by -1 and then adding them to- gether we are getting (ω2 n − ω2 m)XT n mXm = 0 (2.119) Since for n 6= m (ω2 n − ω2 m) 6= 0, XT n mXm = 0 for n 6= m (2.120) If n = m, since (ω2 n − ω2 n) = 0, the product XT n mXn does not have to be equal to zero. Let this product be equal to λ2 n XT n mXn = λ2 n (2.121) Division of the above equation by λ2 n yields ( 1 λn XT n )m( 1 λn Xn) = 1 (2.122) But according to 2.111 1 λn Xn is eigenvector too. Let us denot it by Ξn Ξn = 1 λn Xn (2.123) The process of producing of the eigenvectors Ξn is called normalization and the eigenvector Ξn is called normalized eigenvector or normalized mode. According to 2.122, ΞT n mΞn = 1 (2.124) Taking into account Eq’s 2.120 and 2.124 one can conclude that ΞT n mΞm = ½ 0 if n 6= m 1 if n = m ¾ (2.125) It is said that eigenvectors Ξn and Ξm that fulﬁll the above conditions are orthogonal with respect to the inertia matrix m.

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